Theoretical Basis of Indicator-Dilution Methods
For Measuring Flow and Volume
By Kenneth L. Zierler, M.D.
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• The purpose here is to inquire into the
justification of the use of the indicator-dilution principle for measurement of fluid flow
and volume. What is the validity of the formal expressions? Upon what assumptions are
they based? What are the effects of specified
violations of these assumptions?
An indicator, in the sense used here, is a
substance that permits observations of some
element of volume of the fluid under study.
The indicator shows the position of the element of volume in space and with respect to
time, and distinguishes the indicated element
from all other elements of volume.
In practice, a known quantity of indicator
is introduced into a fluid flowing at unknown
rate through a system of unknown volume.
Fluid is sampled or monitored at one or more
points downstream from the plane of introduction and the concentration of indicator,
diluted by the parent fluid, is measured as a
function of time. Indicator may be introduced
into the system in any of a number of ways,
usually either only once as rapidly as possible
(sudden injection) or continuously at constant rate (constant injection). It is claimed
that from a knowledge only of the quantity
of indicator injected (or of the rate of its
injection for the case of constant injection)
and of the observed concentration of diluted
indicator at the sampling site, over appropriate time intervals, both flow and volume
can be calculated. The validity of this statement has been argued effectively by Stewart,1-4 Hamilton and his colleagues,5 StephenDepartment of Medicine, The Johns Hopkins
University School of Medicine, Baltimore, Maryland.
Original studies described here have been aided
by a contract between the Office of Naval Research,
Department of the Navy, and The Johns Hopkins
University (NE 101-241), and by a grant from the
National Institute of Arthritis and Metabolic Disease (A-750).
Reproduction in whole or in part is permitted for
any purpose of the United States Government.
Circulation Research, Volume X, March 196B
son,6 Sheppard,7 Meier and Zierler,8 and
Burger and colleagues.0 The argument that
follows is adapted from one given previously 8, 10, 11
The Assumptions or the Ideal System
Consider a system with a single inflow and
a single outflow orifice. The internal structure
of the system is of no concern for the present.
Recirculation does not occur; that is, once a
unit of fluid leaves the system it does not reenter. In order to measure flow through and
volume of a system it is necessary that flow
and volume be constant during the period of
measurement. Constant volume implies that
every unit of fluid entering the system must
eventually leave the system. One other stipulation must be made, but this one requires
some exposition. The system must exhibit
stationarity. To understand stationarity we
must look for a moment at a real system, say
a vascular bed.
When an indicator is injected suddenly
into some portion of a vascular bed it does
not all appear suddenly at a sampling site
but it is dispersed with respect to time. The
time required for a given indicator particle
to flow from entrance to exit through the
system, by whatever path, is its transit time.
No one transit time applies to all indicator
particles; rather, there is a family of transit
times. Stationarity demands that the frequency with which each transit time occurs,
that is, the distribution of transit times, must
remain constant during the experiment.
Finally, it is essential that the distribution
of transit times of indicator particles be
identical with the distribution of transit times
of the native fluid; that is, indicator and native fluid must mix thoroughly at the entrance
to the system.
Measurement of Flow and Volume
by Sudden Injection
We now consider measurement of flow and
393
394
ZIERLER
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curve and dividing it into the known quantity of injected indicator.
We now introduce the distribution, or frequency, function, h(t), which describes the
fraction of injected indicator leaving the system per unit time at time, t; that is,
Mt) = Q o(t)/m,
(2)
where Q c(t) is the rate at which indicator
leaves the system at time t.
We must bear in mind that the area under
a frequency function is unity and specifically
t
3
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S
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3
4
time
FIGURE 1
5
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7
o
Concentration of indicator with time following
sudden injection. Plotted as a histogram. (Reprinted from Bulletin of The Johns Hopkins Hospital 103: 199, 195S.)
volume by means of sudden injection of an
indicator. To measure flow, we rely on the
assumption that all the indicator must leave
the system sooner or later.
Let nil units of indicator be injected at time
zero into the entrance to the system, and
measure the concentration of indicator at exit
as a function of time, e(t). The amount of
indicator, dm, leaving the system during a
small time interval between time t and time
t + dt is the concentration of indicator, c(t),
multiplied by the vokime of fluid leaving the
system during this time interval, and this is
the flow, Q, in units of volume/time, multiplied by the time interval, dt; that is, dm =
c(t) Q dt.
Becaiisc all indicator, m,, must leave the
system, nil equals the sum of the amounts
leaving the system during all such time intervals, or
CO
c(t) Q dt =
c(t) dt,
^0
whence
Q=-
f
Jo
Till
(1)
c(t) dt
The unknown flow, Q, through the system
is determined by measuring the area under
the obsei'ved indicator concentration-time
h(t) dt = Q f
c(t) dt/m,
Jo
=
= 1.
Combining equations (1) and ( 2 ) ,
h(t) = c ( t ) /
f
c(t) dt.
(3)
Thus, to determine h(t)
Jo for fluid particles,
each experimentally observed c(t) for indicator is divided by the area under the curve
e(t) versus t.
Consider, for example, a system in which
indicator is diluted so that its concentration
at outflow appears as illustrated in figure 1.
Then, the frequency function, h(t), is as illustrated by the histogram in figure 2, in which
two twelfths of the fluid have transit times
occurring during the third unit of time, four
twelfths of the fluid have transit times during
the fourth time unit, three twelfths during
the fifth, two twelfths during the sixth, and
the remaining one twelfth during the seventh.
To find the volume of fluid present in the
system at time zero, imagine that the particles
of fluid that compose the volume can be distinguished by their transit times. An element
of volume, dV, is made up of all those particles which, initially present at time zero,
have transit times between t and t + dt. The
fraction of particles that require times between t and t + dt to leave is h(t) dt. Some
of these particles have just entered the system at time zero. Others entered the system
t units before time zero and are therefore just
ready to leave the system at time zero. The
rest of the particles making up dV entered
the system during all times between zero and
t units before zero.
Circulation Research, Volume X, March 1962
395
SYMPOSIUM ON INDICATOR-DILUTION TECHNICS
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FIGURE 3
FIGURE 2
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Frequency function of transit times obtained from
figure 1. (Reprinted from Bulletin of The Johns
Hopkins Hospital 103: 199, 1958.)
The rate at which all fluid particles enter
the system and leave it is Q. The rate at which
particles making up dV leave the system is
therefore Q h(t) dt. Some of these particles
leave at time zero, and particles of this transit time continue to leave the system until
time t, at which instant all such particles,
originally present in the system at time zero,
will have been eliminated.
The volume of such particles, dV, is the
time required for them to leave, t, multiplied
by the rate at which they leave, Q h(t) dt, or
dV = t Q h ( t ) dt.
(4)
If, at time zero, indicator is introduced
into the system in such a way as to be mixed
thoroughly with inflowing native fluid (where
mixing is denned by the fact that the distribution of transit times of indicator particles
is the same as that of native fluid), those indicator particles requiring times between t
and t + dt to leave the system can be regarded
as pushing out ahead of them all fluid particles characterized by the same transit time.
The element of volume, dV, can be regarded
as a bubble flowmeter, in which the indicator
is the bubble. When indicator appears at exit,
exactly one element of volume has been
washed out.
The process is illustrated by figures 3, 4
and 5, in which the distribution of transit
times is the same as that shown in figures 1
Circulation
Jiaacarch. Volume X, March 1962
Evolution of indicator-dilution curve for measurement of volume by sudden-injection method. For
explanation see text. See also figures 4 and 5.
(Redrawn from Bulletin of The Johns Hopkins
Hospital 103: 199, 1958.)
and 2. The vertical axis is Q h(t), that is, the
fraction of the flow with transit times between
t and t + dt leaving the system per unit time
at any time. The horizontal axis from left to
right is the transit time characterizing each
element of volume. The horizontal axis projected toward the observer is experimental
time elapsed after addition of indicator to
the inflow.
During the first experimental time unit, no
indicator appears at outflow (fig. 3). The 12
white cubes illustrate that during the first
time unit, native fluid appears at exit with
varying transit times; two twelfths of the
outflow entered the system 3 time units earlier, four twelfths entered 4 time units earlier,
three twelfths entered 5 time units earlier,
and so on. Indicator does not appear until
the third time interval (fig. 4), when all those
particles with the shortest transit time, originally present in the system at time zero, have
now left the system. Indicator particles requiring longer than 3 time units have not .yet
appeared, so that only the fastest two twelfths
of the outflow is marked by the indicator,
illustrated by shaded cubes. However, we
now have information about that element of
volume, dV, characterized by those particles
whose transit, times occupy the third time interval. The volume of this element is the
396
ZIERLER
Because h(t) is the frequency function of
-
transit times,
i&i 4 1i>
/
i
e
PARTICLE
3
6
6
7
5
4
JQ
tion the mean of all transit times or the mean
transit time, t. Therefore
V = Q t,
(6)
which states the fundamental fact that volume equals flow multiplied by mean transit
time.
In terms of the observed concentration of
indicator, from equation (3) and equation
(5),
.
•fii 44 55
111 1
r- °°
1
t h(t) dt is by defini-
TRANSIT TIME
FIGURE 4
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also figures 3 and 5.
t c(t) dt
experimental elapsed time (3 time units) multiplied by Q h(t) (2 units) multiplied by the
time interval dt (in this case, 1 unit, that is,
the third time unit) or t Q h(t) dt equals 6
volume units.
During the fourth time interval, the fraction of the outflow that has transit times during the third time unit is of no interest to us
because it entered the system after indicator
was introduced and was therefore not part
of the volume of the system at time zero. During the fourth time interval all indicator particles with transit times greater than 3 and
as large as 4 time units will appear at the
outflow, so that a second element of volume
will have been displaced from the system.
And so the process continues until all indicator has left the system, in this example, at
the end of the seventh time unit (fig. 5). We
now have five elements of volume, all characterized by their transit times, all marked by
the appearance of indicator at outflow. The
distribution of indicator over the experimental elapsed time is exactly the same as the
distribution of transit times of native fluid
particles at the outflow from the system at
any time.
"We can now determine the total volume of
the system simply by adding together all the
elements of volume in figure 5. Formally, we
integrate equation (4) to find
t h(t) dt.
(5)
(7)
1
c(t) dt
where the ratio of integrals is nothing more
than instructions for finding t.
Measurement of Flow and Volume
by Constant Injection
When indicator is introduced continuously
into the entrance to the system at constant
rate, liii, in units of mass per time, if mixing
at inflow is complete, the concentration of
indicator admitted to the system is iiii/Q
(fig. 3, 4 and 5). The system is unaware of
whether the indicator diluter is using sudden
or constant injection. In the illustrated case,
no indicator appears at exit until the third
time interval when the element of volume
characterized by the briefest transit times
will display the first indicator particles. In
this element of volume the concentration of
indicator is nii/Q during the third time interval. The fraction of outflowing particles
with this transit time is h(t) dt. Therefore,
the concentration of indicator in the total outflow at this time is h(t) dt • riii/Q. Unlike
the case of sudden injection, when indicator
is injected constantly, once it has appeared
at exit in a given element of volume it will
continue to appear, as illustrated in figure 6.
During the next, or fourth, time interval,
indicator is contributed to the outflow by the
element of volume with the next most rapid
transit times, as well as by the element of
Circulation Research, Volume X, March 106t
397
SYMPOSIUM ON INDICATOR-DILUTION TECHNICS
/
e
34
PARTICLE
I
5
TRANSIT TIME
*•
FIGURE 5
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See also figures 3 and 4.
transit times from zero to t is
h(s) ds.
The concentration of indicator at outflow at
time t is therefore
Ms) ds-
<4 J o
(g)
h(t) dt =
We have already seen that
1
Jo
Lim C(t) =
1, therefore
t->co
Q
2
3
TRANSIT
4
5
TIME
6
7
>-
FIGURE 6
Same a<s figure 5, but for constant-injection
od. For explanation see text.
volume that has already contributed indicator. The fraction of outflowing particles with
c(t) = 4 r f
/
PARTICLE
— Cinnx
Equation (9) reminds us that C(t) will increase to a maximal and constant value, and
that we make no new assumptions to be sure
that such is the case. It also of course states
that once C,,,nx is reached, the unknown flow
can be measured:
(10)
Q = »VCm.,x.
The volume of the system can be determined by one of two equivalent methods.
When the concentration of indieator at outflow finally reaches Cmnx = nu/Q, the concentration of indicator everywhere within the
system must also be iiii/Q. This means that
there is within the system a mass of indicatoi',
mv, distributed over V at concentration riii/Q,
or
mv/V = lii./Q = C,n,,x.
Circulation Research, Volum* X, March 1902
meth-
Since Cmnx is measurable, V becomes known
if we can determine mv. mv is determined as
follows:
The amount of indicator in the system at
any time, t, is
m v (t) = (input up to time t) —
(output up to time t).
Input up to time t is simply the known constant rate of injection, mlt multiplied by the
time, t. The output of indicator between time
t and t + dt is the concentration of indicator
at time t, C(t), multiplied by the volume of
fluid leaving the system between t and t + dt,
which is Q dt, or the output is C(t) Q dt. The
output of indicator up to time t is the sum
of all such outputs or
rt
m v (t) = iii,t -
So
I
Q C(s) ds
[in, - Q C(s)] ds
ds
[ C ™ . - C(s)] ds.
The concentration of indicator within the
system at time t is
m v (t)
v
_
Q
y
- C(s)] ds.
398
ZIERLER
The limit of this concentration
have seen, C mnx . Therefore,
Lim
t-»oo V
Whence,
V =
- X fc
is, as we
— O ranx
~~QJ0
H(t) =
I
h(s) ds.
(12)
Jo
L'niax J 0
[ C m n x - C ( t ) ] dt. (11)
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The integral in equation (11) is the area
between the line Cmax, extrapolated back to
zero time, and the curve C(t), that is, it is
the area above the build-up of indicator concentration.
We know from equation (6) that V = Q t
Comparison of equation (6) and equation
(11) states that
I
We introduce the cumulative distribution
function, H ( t ) , which is the integral of the
distribution function, h ( t ) , or
= -±— f
Combining equations (8) and (12),
C
( t ) = -77- H ( t ) ,
C(t)
H(t) =
Cmnx "
Returning to equation (11), which we wish
to identify with equation (6), we substitute
H(t) for'c(t)/C m «x:
or,
V = Q f °° [ l - H ( t ) ] dt.
%J o
fC max -C(t)] dt.
(13)
The problem now is to prove the identity of
It is of interest to prove this identity in another way.
We begin by redeveloping equation (8),
which evolved from inspection of figure 6, in
a way that lends itself to further generalization that will be useful later when we examine
recirculation.
Consider the contribution to the rate at
which indicator leaves the system at time t
made by indicator introduced into the system
during the time interval between s and s + ds
time units before t, that is, in the vicinity of
time t — s. The amount of indicator introduced during this time interval is H'M ds. Of
that indicator, the fraction eliminated per
unit time at time t is h(s). Therefore, of the
indicator, riii ds, introduced between s and
s + ds time units before t, the amount leaving
per unit time at t is h(s) MM ds. Summing
for all such time intervals before t, the rate
at which indicator, injected from time zero
to time t, leaves the system at time t is
t
riii Ms) ds.
But the rate at which indicator leaves the
system is also Q C(t). Therefore,
f
[ 1 - H(t)l dtwitht =
Jo
•^0
th(t) dt.
This is seen b.y integrating by parts.
J"'
Jo
[ l - H ( s ) ] ds = t [ l -
+ I
h(s) ds.
(8)
s h(s) ds
or,
S,
0
[ 1 - H(t) I dt = Lira t [1 - H ( t ) ]
t-> oo
s* 00
t h(t) dt.
+ I
Jo
Figure 7 illustrates the graphic meaning of
the parts of the integral and shows that the
integrals must both be finite or both be infinite. They cannot both be infinite since this
would yield infinite volume. For the case of
finite integrals,
Lira t [1 - H(t)] = 0. Therefore,
t-»oo
. 00
C(t) = - ^ - f
Q JQ
00
J
0
(* 00
[1 - H(t)] dt = I t h(t) dt - f.
JO
(14)
Circulation Research, Volume X, March 196t
399
SYMPOSIUM ON INDICATOR-DILUTION TECHNICS
Relation Between Single and Constant
Injection
Differentiation of equation (8) with respect
1o time yields
d C(t) _ JUJ_
(It
Q h(t).
But for the case of sudden injection, h(t)
— Q e(t)/m|. Combining the above two equations,
d C(t)
d t
=
ifij
in i
c(t)
(15a)
and
C(t) =
nil
mi
c(s) ds.
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(15b)
Equation (15) states that the concentration
at outflow following sudden injection is simply some multiple of the derivative of the
concentration at outflow during constant injection. This means that the time course of
the concentration curve following sudden injection coincides with the time course during
constant injection. The instant some indicator
appears following sudden injection so that
c(t) just assumes a nonzero value, C(t) also
just assumes a nonzero value, that is, appearance time is the same for both. When c(t)
reaches its maximal value, that is, when its
first derivative is zero, C(t) will simultaneously have a flex point. When c(t) returns to
zero, d C (t)/dt becomes zero and C(t) just
reaches its maximal value, Cmax. Irregularities in one curve will appear simultaneously
in the other. For example, when recirculation
is apparent in c(t) it will also be apparent
in C(t). Manipulation of one curve, for example, in an effort to exclude the effect of
recirculation, is as simple or as difficult for
the other curve. The choice between sudden
and constant injection technics lies not in the
formal treatment of the data but in the individual experiment.
Effects of Violation of Assumptions,
or Real Vascular Systems
1. The System Does Not Have a Single Infloiv and a Single Outflow Orifice.—If the
system has either a single inflow or a single
outflow or if somewhere within the system
(here is a single channel through which all
flow must pass and in which mixing occurs
Circulation Research, Volume X, March 196B
FIGURE 7
Graphic representation of the fact that
oo
[_1 - JK(t)] dt is mean transit time
s:
[1 - B(s)2
ds is the sum of the two shaded areas,
rt
of xohich I
s h (s) ds is the loiver.
^0
For the limit of the loiver area to remain finite a>s
t—^ao, the area of the upper rectangle must approach zero, which it does. (Reprinted from Journal of Applied Physiology 6: 731, 1954.)
(as in systems that include the heart, provided it can be shown that mixing of indicator
and blood is complete), then flow can be measured by either sudden or constant injection
by the equations developed earlier, equations
(1) aud (10). This is because equation (1)
depends only on the fact that all indicator
must eventually leave the system and that at
some time between t and t + dt prior to sampling, all indicator has been diluted by a volume equal to Q dt, and because equation (10)
depends only on the fact that once injected
indicator has been diluted so that its concentration is riii/Q it cannot be diluted further
nor can it be concentrated, and outflow concentration must sooner or later become constant at liii/Q.
However, it is not even essential that all
fluid pass through a single channel. If fluid
from each input channel mixes with that from
every other input channel before leaving the
system, it may still be possible to measure
flow. This thesis is developed in reference 11.
The criterion for satisfactory intermingling
is simply that, for the case of constant injection, the limiting concentration of indicator
in every output channel must be the same.
The volume measured hy the equation
400
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V = Q I may or may not have little relation
to some anatomic volume of interest.
Consider many inputs, Ai, A2, A3, and so
on, and a single output, B. Indicator is injected suddenly into one input, say Aj. Let
the flow past Ai be Qi, past A2 be Q2, and
so on, where Qi + Q2 + • • • = Q, the total
flow past B. There is an experimentally determined mean transit time, I. Then there is
a volume beginning at Ai and proceeding to
B such that Vi — Qt i. There is also a volume
beginning at A2 such that V2 = Q2 t, and so
on. Summing all these volumes, the volume
of the system is
V = Vi + V2 + . . . + Vn = Qi t + Q 2 1 +
. . . + Qn I = Q I.
In this case, then, the relation V = Q t is
used to define a volume and it is up to the
investigator to decide whether or not he needs
to identify anatomically the points A2, A3,
and so on defined by the relation given above.
The same argument applies to the case of a
single input, many outputs in which only one
output is sampled. The volume in either of
these cases includes a portion of each input
or each output channel up to that site at
which the mean transit time is the same as
that determined for the injection-to-sampling
site actually used.
2. The System Is Nonstatiovary.—The stationarity condition, that is, that the distribution of transit times for entering particles
does not change with time, is violated in real
vascular systems. It is certainly violated in
pulsatile systems, particularly those that include the heart, and it is apt to be violated
by vasomotor activity. If phasic, not necessarily regular, alterations in distribution of
transit times fluctuate rapidly about some
central value and if the periods of the phases
are brief compared to the time required for
evolution of the sudden-injection indicator
concentration-time curve, then the violation
of stationarity may not be important.
3. Flow or Volume or Both Not Constant.
—The sudden-injection method fails completely in this ease. The constant-injection
method cannot measure a changing volume
but it may measure flow. If the rate at which
flow changes is slow (time constant of the
ZIERLER
change is long compared to mean transit
time) or if the change is from one steady flow
to another, the limiting concentration, iiii/Q,
will reflect altered flow appropriately.
4. Flow of Indicator Particles Is Not Representative of Native Fluid.—Obviously, the
investigator must accept responsibility for
selection of an appropriate indicator. In the
case of a heterogeneous fluid, such as blood,
coursing through a leaky system, such as the
cardiovascular net, it may be difficult to find
aii indicator for which the distribution of
transit times is that of the component of the
native fluid under investigation.
As an example of the difficulty, consider
the case in which the indicator is tagged serum albumin and the native fluid is blood
plasma flowing through some portion of the
vascular system in which there are capillaries.
Insofar as there is net water loss from plasma
at the arteriolar end of capillaries, indicator
protein will be concentrated. For the case of
constant injection, indicator leaves arterioles
at concentration liii/Q. In the proximal portions of capillaries its concentration exceeds
liii/Q. If all water returns to capillaries at
venular ends, concentration of indicator entering venules is again 1H1/Q. If the water
does not return completely to venular ends
of capillaries, as in edema formation, in prominent lymphatic flow or in the elaboration of
urine, concentration of indicator in veins will
exceed riii/Q, where Q is arterial inflow, but
will equal mi divided by venous outflow. The
latter is, therefore, the flow measured.
Even if water simply exchanges across capillaries with no net loss from the vascular system, the methods must underestimate the volume. By the sudden-injection method, the
mean transit time of indicator is less than that
of water that has an extravaseular circuit.
By the constant-injection method, the mass
of indicator in the system is estimated correctly from the equation mv = Q
f
dt,
but it is no longer true that this mv divided
by V is Cmax because sonic portion of mv exists
at a concentration greater than Cmax- Some
Circulation Research. Volume X. March 106$
401
SYMPOSIUM ON INDICATOR-DILUTION TECHNICS
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bound on the error may be calculable if there
is sufficient information about the system.
II: the indicator diffuses out of capillaries
and does not all return to the vascular system
under study, flow cannot be measured by the
sudden-injection method (because tlie true
value of mi is unknown, unless the appropriate quantity of indicator can be measured
by some independent means). Flow can, however, be measured by the constant-injection
method. If indicator escapes from capillaries
at concentration ihi/Q, the measured flow is
the true inflow that originally diluted the
indicator. Otherwise the asymptotic concentration of indicator at outflow will depend
upon the outflow. The volume measured, however, is the volume in which indicator is distributed and therefore exceeds that of plasma.
The mean transit time, t = V/Q, calculated
from transients of the constant-injection
curve is that oi: tlie indicator and not of
plasma.
An interesting error, discussed elsewhere,8
occurs when an indicator that tags plasma is
used to estimate whole-blood flow and volume.
5. liecirculation Occurs.—Recirculation of
indicator is of practical importance when it
occurs before all indicator particles have
completed the first transit. Recireulation is a
nuisance which confuses interpretation of the
primary circulation curve, making it difficult
to calculate Q and V. On the other hand, the
concentration-time curve during recirculation
may give some information about the channels through which recirculation occurs, as in
the ease of shunts.
There are two ways to handle problems
created by recirculation. One is to extend the
treatment of the concentration of indicator
as a function of flow and volume to include
recirculation; the other is to treat formally
the concentration of indicator during the first
circulation and by some means extract from
the over-all concentration-time curve the concentration-time curve that applies only to the
first circulation.
Let us first examine the possibility of including concentration due to recirculation in
our fundamental equations for flow and volume.
Circulation Research, Volume X, March 196B
We extend the second argument by which
equation (8) was developed. The important
equation that follows was first produced by
Stephenson.0
Indicator is introduced into the inflow of
the system at rate nit(t), where the function
is unspecified for the moment. During the
interval s to s + ds time units before t, that
is, in the vicinity of time (t — s), the amount
of indicator introduced is mi(t — s) ds. The
fraction of this amount eliminated per unit
of time at time t, or s time units later, is h(s).
The contribution to the rate at which indicator leaves the system at time t made by indicator introduced between s and s + ds time
units earlier is the product [h(s)] [ni|(t — s)
ds]. Summing for all such time intervals before t, the rate at which indicator leaves at
t
time t is |
[mi(t — s)] h(s) ds. But
Jo
this rate is also equal to Q C(t), where C(t)
is concentration at outflow. Therefore,
•t
m,(t - s) h(s) ds. (16)
C(t) =
Q ^o
The concentration of indicator at input to
the system is C ( (t) = nii(t)/Q, so that equation (16) may be written
C(t) =
I
C,(t - s) h(s) ds.
(17)
With the aid of figure 8 we shall now develop an expression for C,(t) in terms of a
distribution function. Imagine a flow system
shaped like a doughnut. One plane normal to
the axis of flow is selected arbitrarily as the
input. A quantity of indicator, mt, is injected
suddenly into the input. The concentration of
indicator is then monitored constantly at the
input. Indicator is distributed through the
flow system so that the fraction of it making
the complete circuit per unit of time is described by the frequency function g(t).
In the illustration, no indicator has returned until the fifth time interval, and all
of it has completed the first circuit by the end
of the eighth time interval, shown by the first
set of four white bars. We already know that
this event is described by Ci(t) = m{ g(t)/Q,
402
ZIERLER
or, since Ci(t) = ni|g(t)/Q for the initial
circuit,
C,(t)
- m* cg(tt - s) g(s) ds. (IS)
Q Jo
time —»•
FIGURE 8
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Convolution function illustrating re circulation.
For explanation see text. (Reprinted, from, Bulletin of The Johns Hopkins Hospital 103: 199,
1958.)
In this ease, the distribution function, g(t),
is redistributed, or convoluted, upon itself.
Integrals of the type found in equation (IS)
are called convolution integrals. They can be
manipulated through their Laplace transforms, and they are in fact transformable,
although frequently there is insufficient
knowledge about the functions to make their
transforms useful. More practically, they lend
themselves to analysis by computer technics
so that the distribution function may be retrieved from the integral.
Before leaving figure S we should point out
that with each recirculation, indicator particles are dispersed more and more with respect to time, so that as t grows large, Ci(t)
tends to approach the average concentration
in the system, which of course is the mass of
indicator, nii divided by the total volume, VT.
Therefore,
where nii g(0)/Q = Ci(0). Those particles
that had the shortest transit times during the
first circuit re-enter the system and are distributed in accordance with g(t), so that the
first of them emerge during the ninth time
interval, and by the twelfth time interval, as
shown by the second set of four white bars,
they have all completed the second circuit.
The area of the second set of four white bars
is, of course, the same as that of the first white
bar appearing during the fifth time interval.
Meanwhile, those indicator particles that appeared initially during the sixth time interval, that is, those with the second shortest
transit times on the initial circuit, have reentered the system and been distributed in
accordance with g(t), as shown by the first
set of foiir shaded bars. Their contribution
to Ci(t) is additive to that of particles that
entered earlier but that now take paths with
greater than the shortest transit times. The
white bars added to the shaded bars represent
the contribution of indicator particles that
initially traversed the system with the next
longest transit times. The distribution of successive re-entering particles is marked by
alternate white and shaded bars.
By analogy with equation (17),
Lim C,(t) = |
C,(t - s) g(s) ds = m,/V T ,
Jo
(19)
t-»oo
which provides a measure of volume of the
full recirculating system.
Now let us say that we are interested in
obtaining the flow and volume through only
a portion of the reeircuJating system. Stephenson0 was the first to show that this could be
done by the use of two indicators injected
continuously at constant rate.
If h(t) is the frequency function through
the fraction of the system under study and
Co(t) is concentration at output, we already
know that
C,(t) = |
C,(t) =
C, ( t - s ) g(s) ds,
f
Co(t) =
Jo
00
C i ( t - s ) h(s) ds,
and if C, (t) is concentration at input and
g(t) is the frequency function through the
whole recirculating system, we already know
that
rt
Jo
C ( t - s ) g(s) ds.
Circulation Research, Volume X, March
403
SYMPOSIUM ON INDICATOR-DILUTION TECHNICS
Combining the two equations,
t
h(s f
Cu(t) = f| Ms)
C . ( t - s - r ) g(r) drds.
Jo
Jo
In our discussion of figure 8, we saw that,
the concentration function in a recirculating
system behaves like a damped oscillation and
at any point in the system approaches i\\\
t/VT, where U'M is the rate of constant injection. Therefore, the asymptotic behavior of
concentration at outflow is
Lim C,,(t)
I—»oo
Til |
form ae"1", where a is any arbitrarily selected
concentration on the proper part of the down
limb, until it was obviously interrupted by
an increase in concentration attributed to the
first recirculation of indicator. This means
that, after an appropriate time, a plot of the
(t — s) h(s) ds
VT
+ Jo
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rii, t
(20)
s*
r °
00
I
s h(s) ds
h(s) ds - —~
vT J O
^T J 0
C,(t)
=•
mi I
Q
where i is mean transit time through the desired fraction of the system.
Now, if a second indicator is injected at
constant rate into some point of the recirculating system, outside the primary system
whose Q and V are desired, the concentration
of the second indicator at the outflow from
the primary system (the same point at which
Co is measured) will have as its limit for
large t
Lim K 0 (t) = K|(t) = mi i/VT
(22)
t-»oo
where Ko and Ki are concentrations of the
second indicator.
Equations (21) and (22) determine Q and
I, and therefore V. VT, the total volume,
which appears in both equations, is estimated
independently, or simply by the limit equation (19). Thus, recirculation is an essential
part of the scheme and is not corrected for.
Other methods, exploiting the convolution integral, have been discussed11 and some will be
presented elsewhere in this symposium.
We turn now to consider the possibility of
correction for recirculation without actually
measuring it. Hamilton and colleagues5 discovered that, when an indicator was injected
suddenly intravenously in the dog and the
concentration measured in arterial blood, the
down limb of the concentration-time curve
sooner or later fitted an exponential of the
Circulation Research, Volume X, March 19G2
(21)
logarithm of indicator concentration versus
linear time yields a reasonably straight line
until recirculation appears. When recirculation does appear, the logarithmic down slope
is simply extrapolated to very small concentrations. The extrapolated concentration-time
curve is then replotted on linear coordinates
and, with recirculation thus eliminated, the
area and the mean transit time of the primary
curve are calculated.
In specific instances it may be possible to
find other useful approximations and, indeed,
others exist.11
If reeireulation is a late event, no great
error is introduced by any reasonable extrapolation, and it is far simpler to extrapolate.
However, if recirculation occurs early so that
the down limb following sudden injection is
obscured, then no correction for recirculation
can be made with confidence. It is then necessary to use one of the other methods that include recirculation as part of the analysis,
such as the procedure proposed by Stephenson.0 These methods, although complex, have
the important advantage that they make no
assumptions about the form of the distribution function.
Effect of Injection Which Is Neither
Sudden Nor Constant
1. Sudden Injection That Is Not Truly In-
404
ZIERLER
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stantaneotis.—Obviously it is impossible to
inject literally instantaneously. If tbe mean
transit time through the system is long compared to the mean time of injection it is usual]}' sufficiently accurate to ignore the fact
that the injection is not really instantaneous.
However, if the mean time of the distribution
of the injection process is significant compared to the mean transit time through the
system, then the equations for measurement
of volume following sudden injection do not
hold because the mean transit time determined from the measured output concentration will exceed the true mean transit time
through the system, described by h(t). The
correct equation is the convolution,
C(t)
s) h(s) ds,
Q Jo
(23)
where hi(t) is the fraction of injectate entering the system per unit of time at time t, that
is, it is the frequency function of injection.
Calculation of flow from the area under the
concentration-time curve is still correct because, as was stated previously,
•» CO
s* t
I
I
> 00
o
/-» 0 0
h,(t) d t = I
Jo
h ( t ) dt = 1,
so that it is still true that
•»
Q = m,/ I
00
C(t) dt.
To calculate the true mean time,
f
p 00
the system, I
t h(t) dt, we determine the
mean time described by the measured conoontration of indicator at outflow ,
f\
C
I
t C(t)
J()
00
dt/ (
Jo
C(t) dt, and subtract from it the
mean time of the injection process,
r
thj(t) dt, which must be measured inde-
If the injection is very rapid, it
pendently.
Jo
may be sufficiently accurate to consider the
mid-point in time of injection as its mean
time.
2. Other Forms of Injection.—These are
not used widely but may have some theoretical interest or practical value in special cases.
The case of injection at constant acceleration
yields a very simple method for determination
of mean transit time.11
Effect of Collecting Catheter
h,(t — s) h(s) ds dt
Jo Jo
J
Therefore, to find the mean time through
t h(t) dt,
advantage is taken
Jo of an important property
of frequency functions. Given the frequency
function
_j
fi(t) = (
f 2 ( t - s ) f3(s) ds,
Jo
then ti = t2 + t8, where tx is mean transit
time for distribution fi(t), that is,
= \ oo
Joo t fi(t) dt,
l2 is mean transit time for f2, and t 3 is mean
time for fa. Tims, mean transit, times are additive.
If blood is led from the system under stud}'
through a device such as a catheter, the distribution of transit times through the system, h(t), is convoluted upon the distribution
of transit times through the catheter, h o (t).
If the sudden-injection technic is used and
the injection itself is sufficiently rapid to
neglect its mean time, then the observed concentration at outflow from the catheter is
C(t)
=
in, f
h(t - s) ho(s) ds, (24)
Q Jo
Jo
which is identical to the case in equation (23)
in which an inflow distribution is convoluted
through h(t), because the sequence in -which
the convolution occurs makes no difference,
that is,
rt
\
Jo
fi(t - s) f2(s) ds
"Jo'0 t f (t - s )
2
fj(s) ds.
Therefore, all the arguments used with reference to convolution of injection upon the
Circulation Research, Volume X, March JUGS
SYMPOSIUM ON INDICATOR-DILUTION TECHNICS
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distribution function, h(t), apply equally to
convolution of the outflow through the catheter upon h(t). Flow is measured correctly
from the usual equation, and the true mean
transit time through the system (excluding
the catheter) is the difference between the
over-all mean transit time (determined from
the observed concentration-time curve) and
the mean time through the catheter, measured
independently by some in vitro method from
the relation t = V/Q, where V is catheter
volume and Q is some experimentally produced flow.
Several investigators have examined the
problem of the effect of the catheter on the
shape of the indicator concentration-time
curve. For calculation of flow we are concerned only •with the area of the curve. For
calculation of volume we need only the mean
transit time and the flow. The shape of the
outflow curve is immaterial.
Formal Expressions for the Distribution
Function
:
" In the previous sections we have not cared
"what form the distribution function of transit
times, h(t), might take. It has been simplest
and completely accurate to let the flow system
determine the function for us by delivering
an indicator at a concentration that is measurable experimentally as a function of time.
However, it would be nice to know whether
h(t) could be placed in some other form in
which it might be operated upon if occasion
demanded. There are two ways to approach
the problem. One is to examine the concentration-time curves obtained experimentally and,
by curve fitting or otherwise, derive an empirical expression for concentration as a
function of time or some other useful relation. The other is to assume that certain given
laws govern the distribution, predict the distribution function from these laws and test
the observed concentration-time curve for
goodness of fit.
1. Empirical Expressions.—The earliest of
these is attributable to Allen and Taylor12
who treated the sudden-injection indicator
concentration-time curve as a triangle. Hamilton's extrapolation of the down limb as au
Circulation Research, Volume X, March 4?(jg
405
exponential, to which reference has already
been made, was of course also an empirical fit,
although it has since formed the basis for development of a model system.
Several investigators have asked whether
or not indicator-dilution curves commit themselves as soon as they have written the rising
limb and reached peak concentration. To this
end Dow13 examined a large number of curves
obtained by sudden injection into the cardiopulmonary circuit and obtained an expression
for the area under the curve (which equals
nii/Q) from a knowledge only of the time at
which indicator just appeared at outflow (the
shortest transit time or appearance time), the
time at which peak concentration was reached
(the time at which the mode of the distribution function occurred) and the peak concentration. Keys and associates14 found that, in
man, Dow's formula systematically underestimated flow and they applied a correction
factor.
A closely related approach was taken by
Hetzel and colleagues15 who compared the
area of what they called the forward triangle
to the total area of the curve corrected for
reeirculation. The forward triangle is the area
under the rising concentration curve, from
appearance time to time of peak concentration (modal time) following sudden injection.
These methods have been reviewed in
greater detail elsewhere.11 Their development
was stimulated by the need to handle the problem of early appearance of recirculation. The
empirical correlations were prepared from
curves in which recirculation was necessarily
late, so that it could be eliminated by Hamilton's exponential extrapolation. There is no
information as to whether or not correlations
from the one set of experiments, from which
the constants of the equations were obtained,
can be applied to other sets of experiments.
Furthermore, these empirical formulas yield
approximations only of the area of the curve.
They can be used only for estimates of flow,
not of volume.
2. Theoretical Approaches.—"When a Newtonian fluid flows through a long straight tube,
of uniform bore, providing the Eoynolds number is below a certain critical yuluo, its be-
406
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havior is described as laminar and it obeys
Poiseuille's law. Several treatments of a system in which indicator is distributed through
a laminar flow system have been given.16"18
Of course, the resulting curves do not in
themselves apply to a branching vascular system, although verisimilitude is improved by
connecting laminar flow systems in series and
in parallel.11
Sheppard and Savage10 suggested a probabilistic "random walk" approach to the question of curve shape. From the empirical
point of view it was noted that the relation
was approximated by a normal distribution
if a log scale was used on the horizontal rather
than the vertical axis. Stow and Hetzel20 reported a reasonable fit. Although such curves
do arise in the theory of "random alms,"21
the agreement seems purely fortuitous so far
as basic theoretical significance may be concerned. Skewed probability distributions
often become more like the normal after a log
transformation of the abscissas. The down
limb of the normal error curve falls more rapidly than Hamilton's simple exponential, so
that the two methods do not yield the same
flow.
Sheppard7'21 later extended the randomwalk concept and indicated its validity in relation to experiments with dye in long glassbead columns. Vascular beds are considered
to be randomizing nets, but the random progress of dye in a more general sense may also
be included in the concept. Experiments were
performed on glass-bead systems contained in
spherical flasks. Here, where the system deviated to a great extent from the ideal longbead column, a small but definite amount of
horizontal shifting became necessary in obtaining a satisfactory fit.
Another specific form, suggested originally
by Hamilton and his collegues,5 was adopted
by Newman and his colleagues.22 The exponential down limb of Hamilton and his colleagues suggested that indicator was washed
out of a chamber in which it had been mixed
thoroughly. The general equations for such a
system were given by Sheppard.21 Newman
and co-workers22 claimed that the exponential
washout was Q/V and that it could therefore
ZIERLER
be used to measure volume. In general, this
has not been correct; at least the magnitude
of whatever has been determined has not
agreed with volume determined from the relation V = Q I. It will be noted that if the
slope is Q/V it must be t. It has been shown
elsewhere11 that this cannot be the case. Newman and colleagues 22 have also considered
the effect of placing mixing chambers in series. I obtain a slightly different equation,
by plugging exponential functions into convolution integrals, owing chiefly to having set
different boundary conditions,11 but I conclude that in general the slope of the falling
concentration curve cannot be used to measure volume.
If indicator distribution is limited to the
chambers of the heart, then Newman's model
becomes more plausible, and may do very well
as a first approximation despite evidence that
mixing of blood is indeed not complete and
immediate in the right ventricle. The equations are step functions, owing to the discrete
nature of cardiac systole and diastole. They
assume that there is some volume or volumes
in the cai-diac chambers in which mixing of
indicator and blood is complete and instantaneous.
For the simplest case, consider that an
amount of indicator, m,, is introduced into a
ventricle during diastole. Bnd-diastolie volume, with which indicator is assumed to be
mixed, is Vrt. During systole a quantity of
blood, Vs is ejected forward and there is no
regurgitation. At the end of systole there remains in the ventricle a volume V,. = V(i — Vs.
During diastole a volume equal to Va, containing no indicator, flows from atrium to
ventricle. From these assumptions it is easy
to show that
c(t) =
"f (^r) tF>
where F is beats per unit of time (heart rate),
and where for n F < t < (n + 1) F and n
is an integer, t assumes the value of n. Since
flow can be determined by the usual equations, and so can Vd, the residual volume can
be estimated from the above equation.
Because flow equals stroke volume multiCirculation Research, Volume X, March 196S
407
SYMPOSIUM ON INDICATOR-DILUTION TECHNICS
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plied by heart rate, or Q = F Vs, F = Q/Vs
= l./t; that is, heart rate is the reciprocal of
the mean transit of the cardiac output
through a volume equal to that ejected with
each systole.
More complicated systems have been examined. A solution for a two-chambered system,
including regurgitant flow, is given by
McClure and colleagues.23
With regard to all of the various formal
expressions for the distribution function of
transit times, it, is not likely that any arbitrary distribution function will describe the
real distribution function as well as the properly obtained indicator concentration-time
curve.
References
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3. STEWART, G. N.: The output of the heart in
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4. STEWART, G. N.: The pulmonary circulation
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KINSMAN, J. M., MOORE, J. W., AND HAMILTON,
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6. STEPUENSON, J. IJ.: Theory of the measurement
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HETZEL,
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Theoretical Basis of Indicator-Dilution Methods For Measuring Flow and Volume
Kenneth L. Zierler
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Circ Res. 1962;10:393-407
doi: 10.1161/01.RES.10.3.393
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